Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 3

Home , Cyclic permutation

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7 Beyond Hopf maps: even higher dimensional generalization 26 7.1 Clifford algebra: another generalization of complex numbers...... 26 7.2 Mathematical aspects of fuzzy sphere ...... 27 7.3 Hopf spinor matrix and SO(2k)monopole...... 28 7.4 SO(2k +1)Landaumodel...... 30 7.5 Dimensionalhierarchy ...... 31

8 Summary and discussions 32

A 0th Hopf map and SO(2) “Landau model” 33 A.1 Realization of the 0th Hopf map ...... 33 A.2 SO(2)“Landaumodel” ...... 34

B Generalized 1st Hopf map and SU(k + 1) Landau model 35 B.1 SU(k + 1) Landau model and fuzzy complex projective space ...... 35 B.2 Relations to spherical Landau models ...... 37

References 38

1 Introduction

Fuzzy two-sphere introduced by Madore [1] was a typical and important fuzzy manifold where particular notions of fuzzy geometry have been cultivated. In subsequent explorations of fuzzy geometry, higher dimensional and supersymmetric generalizations of the fuzzy spheres were launched by Grosse et al. [2, 3, 4]. Furthermore, as well recognized now, string theory provides a natural set-up for non-commutative geometry and non-anti-commutative geometry [5, 6, 7]. In particular, classical solutions of matrix models with Chern–Simons term are identified with fuzzy two-spheres [8], fuzzy four-spheres [9] and fuzzy superspheres [10]. Fuzzy manifolds also naturally appear in the context of intersections of D-branes [11, 12, 13]. In addition to applications to physics, the fuzzy spheres themselves have intriguing mathematical structures. As Ho and Ramgoolam showed in [14] and subsequently in [15] Kimura investigated, the commutative limit of (even-dimensional) fuzzy sphere takes the particular form1:

S2k SO(2k + 1)/U(k). F ≃ 2k From this coset representation, one may find, though SF is called 2k-dimensional fuzzy sphere, its genuine dimension is not 2k but k(k + 1). Furthermore, the fuzzy spheres can be expressed as the lower dimensional fuzzy sphere-fibration over the sphere

S2k S2k S2k−2. F ≈ × F 2k 2k−2 Thus, SF has “extra-dimensions” coming from the lower dimensional fuzzy sphere SF . One may wonder why fuzzy spheres have such extra-dimensions. A mathematical explanation may go as follows. Fuzzification is performed by replacing Poisson bracket with commutator on a manifold. Then, to fuzzificate a manifold, the manifold has to have a symplectic structure to be capable to define Poisson bracket. However, unfortunately, higher dimensional spheres S2k (k 2) do not accommodate symplectic structure, and their fuzzification is not straightforward. ≥ A possible resolution is to adopt a minimally extended symplectic manifold with spherical symmetry. The coset SO(2k + 1)/U(k) suffices for the requirement.

1 2k−1 The odd-dimensional fuzzy spheres can be given by SF ≃ SO(2k)/(U(1) ⊗ U(k − 1)) [16, 17]. For instance, 3 2 2 SF ≈ S × S . Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 3

Since detail mathematical treatments of fuzzy geometry have already been found in the excellent reviews [18, 19, 20, 21, 22, 23], in this paper, we mainly focus on a physical approach for understanding of fuzzy spheres, brought by the developments of higher dimensional quantum Hall effect (see as a review [24] and references therein). The lowest Landau level (LLL) provides a nice way for physical understanding of fuzzy geometry (see for instance [25]). As we shall see in the context, generalization of the fuzzy two-spheres is closely related to the generalization of the complex numbers in 19th century. Quaternions discovered by W. Hamilton were the first generalization of complex numbers [26]. Soon after the discovery, the other division algebra known as octonions was also found (see for instance [27]). Interestingly, the division algebras are closely related to topological maps from sphere to sphere in different dimensions, i.e. the Hopf maps [28, 29]. While the division algebras consist of complex numbers C, quaternions H and octonions O (except for real numbers R), there is another generalization of complex numbers and quaternions; the celebrated Clifford algebra invented by W. Clifford [30]. The generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-sphere to its higher dimensional cousins. Particular geometry of fuzzy spheres directly reflects features of Clifford algebra. With non-Abelian monopoles in generic even dimensional space, we explain the particular geometry of fuzzy spheres in view of the lowest Landau level physics. There is another important algebra invented by H. Grassmann [31]. Though less well known compared to the original three Hopf maps, there also exists a graded version of the (1st) Hopf map [32]. We also discuss its relation to fuzzy supersphere. The organization is as follows. In Section 2, we introduce the division algebras and the Hopf maps. In Section 3, with the explicit construction of the 1st Hopf map, we analyze the Landau 2 problem on a two-sphere and discuss its relations to fuzzy two-sphere SF . The graded version of the Hopf map and its relation to fuzzy superspheres are investigated in Section 4. In Section 5, 4 we extend the former discussions to the 2nd Hopf map and fuzzy four-sphere SF . In Section 6, we consider the 3rd Hopf map and the corresponding fuzzy manifolds. In Section 7, we generalize the observations to even higher dimensional fuzzy spheres based on Clifford algebra. Section 8 is devoted to summary and discussions. In Appendix A, for completeness, we introduce the 0th Hopf map and related “Landau problem” on a circle. In Appendix B, the SU(k + 1) Landau model and fuzzy CP k manifold are surveyed.

2 Hopf maps and division algebras

As the division algebras consist of only three algebras, there exist three corresponding Hopf maps, 1st, 2nd and 3rd. The three Hopf maps

S1 S3 S2 (1st) −→ S7 S4 (2nd) −→ S15 S8 (3rd) −→ are closely related bundle structures of U(1), SU(2), and SO(8) monopoles [33, 34, 35]. Interes- tingly, the Hopf maps exhibit a hierarchical structure. Each of the Hopf maps can be understood as a map from a circle in 2D division algebra space to corresponding projective space:

1 1 S 1 SC CP (1st) 1 −→1 SH HP (2nd) 1 −→1 SO OP (3rd) −→ 1 For instance, in the 1st Hopf map, the total space SC represents a circle in 2D complex space, i.e. S3, and the basespace CP 1 denotes the complex projective space equivalent to S2. For the 2nd and 3rd Hopf maps, same interpretations hold by replacing complex numbers C with 4 K. Hasebe quaternions H, and octonions O, respectively. For later convenience, we summarize the basic properties of quaternions and octonions. The quaternion basis elements, 1, q1, q2, q3, are defined so as to satisfy the algebra [26]

2 2 2 q = q = q = q q q = 1, qiqj = qjqi (i = j), 1 2 3 1 2 3 − − 6 or equivalently,

qi,qj = 2δij, [qi,qj] = 2ǫijkqk, { } − where ǫijk is Levi-Civita antisymmetric tensor with ǫ123 = 1. Thus, quaternion algebra is noncommutative. As is well known, the quaternion algebra is satisfied by the Pauli matrices with the identification, qi = iσi. (We will revisit this point in Section 5.) An arbitrary quaternion − is expanded by the quaternion basis elements:

3 q = r01+ riqi, i X=1 where r0, ri are real expansion coefficients, and the conjugation of q is given by

3 ∗ q = r 1 riqi. 0 − i X=1 The norm of quaternion, q , is given by || || 3 ∗ ∗ 2 2 q = q q = qq = r + ri . || || v 0 u i=1 p p u X t It is noted that q and q∗ are commutative. The normalized quaternionic space corresponds to S3, 7 1 and the total manifold of the 2nd Hopf map, S , is expressed as the quaternionic circle, SH:

3 3 ∗ ′∗ ′ 2 2 ′ 2 ′2 q q + q q = r0 + ri + r0 + ri = 1. i i X=1 X=1 Similarly, the octonion basis elements, 1, e1, e2,...,e7, are defined so as to satisfy the algebras

eI , eJ = 2δIJ , [eI , eJ ] = 2fIJKeK , (2.1) { } − where I, J, K = 1, 2,..., 7. δIJ denotes Kronecker delta symbol and fIJK does the antisymmetric structure constants of octonions (see Table 1). The octonions do not respect the associativity as well as the commutativity. (The non- associativity can be read from Table 1, for instance, (e e )e = e = e (e e ).) Due to their 1 2 4 7 − 1 2 4 non-associativity, octonions cannot be represented by matrices unlike quaternions. However, the conjugation and magnitude of octonion can be similarly defined as those of quaternion, simply replacing the role of the imaginary quaternion basis elements qi with the imaginary octonion basis elements eI . An arbitrary octonion is given by

7 o = r01+ rI eI , I X=1 with real expansion coefficients r0, rI , and its conjugation is

7 ∗ o = r 1 rI eI . 0 − I X=1 Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 5

Table 1. The structure constants of the octonion algebra. For instance, f145 = 1 can be read from e1e4 = e5. The octonion structure constants are given by f123 = f145 = f176 = f246 = f527 = f374 = f365 = 1, and other non-zero octonion structure constants are obtained by the cyclic permutation of the indices. 1 e1 e2 e3 e4 e5 e6 e7

1 1 e1 e2 e3 e4 e5 e6 e7 e e 1 e e e e e e 1 1 − 3 − 2 5 − 4 − 7 6 e e e 1 e e e e e 2 2 − 3 − 1 6 7 − 4 − 5 e e e e 1 e e e e 3 2 2 − 1 − 7 − 6 5 − 4 e e e e e 1 e e e 4 4 − 5 − 6 − 7 − 1 2 3 e e e e e e 1 e e 5 5 − 4 − 7 6 − 1 − − 3 2 e e e e e e e 1 e 6 6 7 4 − 5 − 2 3 − − 1 e e e e e e e e 1 7 7 − 6 5 4 − 3 − 2 1 −

The norm of octonion, o , is || || 7 ∗ ∗ 2 2 o = √o o = √oo = r + rI . || || v 0 u I=1 u X t Like the case of quaternion, o and o∗ are commutative. The normalized octonion o∗o = 1 represents S7, and the total manifold of the 3rd Hopf map, S15, is given by the octonionic 1 circle SO,

7 7 ∗ ′∗ ′ 2 2 ′ 2 ′ 2 o o + o o = r0 + rI + r0 + rI = 1. I I X=1 X=1 One may wonder there might exist even higher dimensional generalizations. Indeed, following to the Cayley–Dickson construction [27], it is possible to construct new species of numbers. Next to the octonions, sedenions consisting of 16 basis elements can be constructed. However, the sedenions do not even respect the alternativity, and hence the multiplication law of norms does not hold: x y = x y . Then, the usual concept of “length” does not even exist in || || || || 6 || · || sedenions, and a sphere cannot be defined with sedenions and hence the corresponding Hopf maps either. Consequently, there only exit three division algebras and corresponding three Hopf maps.

3 1st Hopf map and fuzzy two-sphere

In this section, we give a realization of the 1st Hopf map

S1 S3 S2, −→ and discuss basic procedure of fuzzification of sphere in LLL.

3.1 1st Hopf map and U(1) monopole The 1st Hopf map can explicitly be constructed as follows. We first introduce a normalized complex two-spinor φ φ = 1 φ 2 6 K. Hasebe satisfying φ†φ = 1. Thus φ, which we call the 1st Hopf spinor, represents the coordinate on the total space S3, and plays a primary role in the fuzzification of sphere as we shall see below. With the Pauli matrices 0 1 0 i 1 0 σ = , σ = − , σ = , (3.1) 1 1 0 2 i 0 3 0 1 − the first Hopf map is realized as

† φ xi = φ σiφ. (3.2) → 2 It is straightforward to check that xi satisfy the condition for S :

† 2 xixi = (φ φ) = 1. (3.3)

Thus, (3.2) demonstrates the 1st Hopf map. The analytic form of the Hopf spinor except for the south pole is given by

1 1+ x φ = 3 , (3.4) x + ix 2(1 + x3) 1 2 and the correspondingp fibre-connection is derived as

† A = dxiAi = iφ dφ, (3.5) − where 1 Ai = ǫij3xj. −2(1 + x3) The curvature is given by 1 Fij = ∂iAj ∂jAi = ǫijkxk, (3.6) − 2 which corresponds to the field strength of Dirac monopole with minimum charge (see for instance [36]). The analytic form of the Hopf spinor except for the north pole is given by

1 x ix φ′ = 1 − 2 . 2(1 x ) 1 x3 − 3 − The correspondingp gauge field is

′ ′† ′ ′ A = iφ dφ = dxiA , − i where 1 A′ = ǫ x . i 2(1 x ) ij3 j − 3 ′ ′ ′ The field strength F = ∂iA ∂jA is same as Fij (3.6), which suggests the two expressions of ij j − i the Hopf spinor are related by gauge transformation. Indeed,

φ′ = φ g = g φ, · · where g is a U(1) gauge group element

−iχ 1 g = e = (x1 ix2). √1 x 2 − − 3 Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 7

Here, the gauge parameter χ is given by tan(χ)= x2 . The U(1) phase factor is canceled in the x1 map (3.2), and there always exists such U(1) gauge degree of freedom in expression of the Hopf spinor. With the formula

∗ 1 ig dg = ǫij xidxj, − 1 x 2 3 − 3 the above gauge fields are represented as i i A = (1 x )g∗dg, A′ = (1 + x )g∗dg, 2 − 3 −2 3 and related by the U(1) gauge transformation

A′ = A ig∗dg. − The non-trivial bundle structure of U(1) monopole on S2 is guaranteed by the homotopy theorem

π (U(1)) Z, 1 ≃ specified by the 1st Chern number 1 c1 = F. 4π 2 ZS 3.2 SO(3) Landau model Here, we consider a Landau model on a two-sphere in U(1) monopole background. In 3D space, the Landau Hamiltonian is given by 1 1 ∂2 1 ∂ 1 H = D2 = + Λ2, (3.7) −2M i −2M ∂r2 − Mr ∂r 2Mr2 i where Di = ∂i + iAi, r = √xixi, and i = 1, 2, 3 are summed over. Λi is the covariant angular momentum Λi = iǫijkxjDk. The monopole gauge field has the form − I Ai = ǫij3xj, −2r(r + x3) and the corresponding field strength is I F = ǫ ∂ A = x . i ijk j k 2r3 i

The covariant angular momentum Λi does not satisfy the SU(2) algebra, but satisfies

2 [Λi, Λj]= iǫijk Λk r Fk . − The conserved SU(2) angular momentum is constructed as

2 Li = Λi + r Fi, which satisfies the genuine SU(2) algebra

[Li,Lj]= iǫijkLk.

On a two-sphere, the Hamiltonian (3.7) is reduced to the SO(3) Landau model [37] 1 H = Λ2, (3.8) 2MR2 i 8 K. Hasebe where R is the radius of two-sphere. The SO(3) Landau Hamiltonian can be rewritten as

1 I2 H = L2 , (3.9) 2MR2 i − 4 where we used the orthogonality between the covariant angular momentum and the field strength; ΛiFi = FiΛi = 0. Thus, the Hamiltonian is represented by the SU(2) Casimir operator, and the eigenvalue problem is boiled down to the problem of obtaining the irreducible representation of SU(2). Such irreducible representations are given by monopole harmonics [38]. The 2 I SU(2) Casimir operator takes the eigenvalues Li = j(j + 1) with j = 2 + n (n = 0, 1, 2,... ). (The minimum of j is not zero but a finite value I/2 due to the existence of the field angular momentum of monopole.) Then, the eigenenergies are derived as 1 I E = n2 + n(I +1)+ . (3.10) n 2MR2 2 In the thermodynamic limit, R,I , with B = I/2R2 fixed, (3.10) reproduces the usual → ∞ Landau levels on a plane: 1 En ω n + , → 2 where ω = B/M is the cyclotron frequency. The degeneracy of the nth Landau level is given by

d(n) = 2l +1=2n + I + 1. (3.11)

In particular, in the LLL (n = 0), the degeneracy is

dLLL = I + 1.

The monopole harmonics in LLL is simply constructed by taking symmetric products of the components of the 1st Hopf spinor

(m1,m2) I! m1 m2 φLLL = φ1 φ2 , (3.12) rm1!m2! where m + m = I (m ,m 0). In the LLL, the kinetic term of the covariant angular 1 2 1 2 ≥ momentum is quenched, and the SU(2) total angular momentum is reduced to the monopole field strength

2 I Li r Fi = xi. → 2R

Then in LLL, the coordinates xi are regarded as the operator

Xi = αLi, which satisfies the definition algebra of fuzzy two-sphere

[Xi, Xj]= iαǫijkXk, (3.13) with α = 2R/I. We reconsider the LLL physics with Lagrange formalism. In Lagrange formalism, importance of the Hopf map becomes more transparent. The present one-particle Lagrangian on a two-sphere is given by m L = x˙ x˙ +x ˙ A , 2 i i i i Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 9 with the constraint

2 xixi = R . (3.14)

In the LLL, the Lagrangian is represented only by the interaction term

LLLL =x ˙ iAi.

From (3.5), the LLL Lagrangian can be simply rewritten as

d L = iIφ† φ, (3.15) LLL − dt and the constraint (3.14) is

φ†φ = 1. (3.16)

It is noted that, in the LLL, the kinetic term drops and only the first order time derivative term survives. The Lagrangian and the constraint can be represented in terms of the Hopf spinor. Usually, in the LLL, the quantization is preformed by regarding the Hopf spinor as fundamental quantity, and the canonical quantization condition is imposed not on the original coordinate on two-sphere, but on the Hopf spinor. We follow such quantization procedure to fuzzificate two-sphere. From (3.15), the conjugate momentum is derived as π = iIφ∗, and the canonical − quantization is given by

∗ 1 [φα, φ ]= δαβ. (3.17) β −I

This quantization may remind the quantization procedure of the spinor field theory, but readers should not be confused: The present quantization is for one-particle mechanics, and the spinor is quantized as “boson” (3.17). Then, the complex conjugation is regarded as the derivative

1 ∂ φ∗ = , (3.18) I ∂φ and the constraint (3.16) is considered as a condition on the LLL basis

∂ φt φ = Iφ . ∂φ LLL LLL

The previously derived LLL basis (3.12) indeed satisfies the condition. By inserting the expression (3.18) to the Hopf map, we find that xi are regarded as coordinates on fuzzy two-sphere

α ∂ X = Rφ†σ φ = φtσ , i i 2 i ∂φ which satisfies the algebra (3.13). Thus, also in the Lagrange formalism, we arrive at the fuzzy two-sphere algebra in LLL. The crucial role of the Hopf spinor is transparent in the Lagrange formalism: The Hopf spinor is first fuzzificated (3.17), and subsequently the coordinates on two-sphere are fuzzificated. This is the basic fuzzification mechanics of sphere in the context of the Hopf map. 10 K. Hasebe

3.3 Fuzzy two-sphere The fuzzy two-sphere is a fuzzy manifold whose coordinates satisfy the SU(2) algebraic relation [1]

[Xˆi, Xˆj]= iαǫijkXˆk.

The magnitude of fuzzy sphere is specified by the dimension of corresponding SU(2) irreducible representation. A convenient way to deal with the irreducible representation is to adopt the Schwinger boson formalism, in which the SU(2) operators are given by α Xˆ = φˆ†σ φ.ˆ i 2 i

t Here, φˆ = (φˆ1, φˆ2) stands for a Schwinger boson operator that satisfy

ˆ ˆ† [φα, φβ]= δαβ, with α, β = 1, 2. Square of the radius of a fuzzy two-sphere reads as

α2 Xˆ Xˆ = φˆ†φˆ)(φˆ†φˆ + 2 . (3.19) i i 4 The commutative expression (3.3) and the fuzzy expression (3.19) merely differ by the “ground state energy”. Thus, the radius of the fuzzy sphere is α R = I(I + 2), I 2 p where I is the integer eigenvalue of the number operator Iˆ φˆ†φˆ = φˆ† φˆ +φˆ†φˆ . In the classical ≡ 1 1 2 2 limit I , the eigenvalue reproduces the radius of commutative sphere →∞ α α RI = I(I + 2) I = R. 2 → 2 p The corresponding SU(2) irreducible representation is constructed as

1 ˆm1 ˆm2 m1,m2 = φ1 φ2 0 , (3.20) | i √m1!m2! | i where m + m = I (m ,m 0), and the degeneracy is given by d(I) = I + 1. Appar- 1 2 1 2 ≥ ently, there is one-to-one correspondence between the LLL monopole harmonics (3.12) and the states on fuzzy sphere (3.20). Their “difference” is superficial, coming from the corresponding representations: Schwinger boson representation for fuzzy two-sphere, while the SU(2) coherent representation for LLL physics. This is the basic observation of equivalence between fuzzy geometry and LLL physics.

4 Graded Hopf map and fuzzy supersphere

Before proceeding to the 2nd Hopf map, in this section, we discuss how the relations between the Hopf map and fuzzy sphere are generalized with introducing the Grassmann numbers, mainly based on Hasebe and Kimura [39]. The Grassmann numbers, ηa, are anticommuting numbers [31]

ηaηb = ηbηa. (4.1) − Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 11

2 In particular, ηa = 0 (no sum for a). With Grassmann numbers, the graded Hopf map was introduced by Landi et al. [32, 40, 41],

S1 S3|2 S2|2, (4.2) −→ where the number on the left hand side of the slash stands for the number of bosonic (Grass- mann even) coordinates and the right hand side of the slash does for the number of fermionic (Grassmann odd) coordinates. For instance, S2|2 signifies a supersphere with two bosonic and two fermionic coordinates. The bosonic part of the graded Hopf map (4.2) is equivalent to the 1st Hopf map.

4.1 Graded Hopf map and supermonopole As the 1st Hopf map was realized by sandwiching Pauli matrices between two-component normalized spinors, the graded Hopf map is realized by doing UOSp(1 2) matrices between | three-component (super)spinors. First, let us begin with the introduction of basic properties of UOSp(1 2) algebra [42, 43, 44]. The UOSp(1 2) algebra consists of three bosonic generators Li | | (i = 1, 2, 3) and two fermionic generators Lα (α = θ1,θ2) that satisfy 1 1 [Li,Lj]= iǫijkLk, [Li,Lα]= (σi)βαLβ, Lα,Lβ = (ǫσi)αβLi, (4.3) 2 { } 2 where ǫ = iσ . As realized in the first algebra of (4.3), the UOSp(1 2) algebra contains the 2 | SU(2) as its subalgebra. Li transforms as SU(2) vector and Lα does as SU(2) spinor. The Casimir operator for UOSp(1 2) is constructed as |

C = LiLi + ǫαβLαLβ,

1 and its eigenvalues are given by L(L + 2 ) with L = 0, 1/2, 1, 3/2, 2,... . The dimension of the corresponding irreducible representation is 4L + 1. The fundamental representation matrix is given by the following 3 3 matrices ×

1 σi 0 1 0 τα li = , lα = t , 2 0 0 2 (ǫτα) 0 − t t where σi are the Pauli matrices, τ1 = (1, 0) , and τ2 = (0, 1) . With li and lα, the graded Hopf map (4.2) is realized as

‡ ‡ ϕ xi = 2ϕ liϕ, θα = 2ϕ lαϕ, (4.4) → where ϕ is a normalized three-component superspinor which we call the Hopf superspinor:

ϕ1 ϕ = ϕ . (4.5)  2 η   Here, the first two components ϕ1 and ϕ2 are Grassmann even quantities, while the last component η is Grassmann odd quantity. The superadjoint is defined as2 ‡ ϕ‡ = (ϕ∗, ϕ∗, η∗), 1 2 − 2 ∗ ∗ ∗ ∗ ∗ The symbol ∗ represents the pseudo-conjugation that acts as (η ) = −η, (η1η2) = η1 η2 for Grassmann odd quantities η1 and η2. See [45] for more details. 12 K. Hasebe and ϕ represents coordinates on S3|2, subject to the normalization condition

ϕ‡ϕ = ϕ∗ϕ + ϕ∗ϕ η∗η = 1. 1 1 2 2 − 2|2 xi and θα given by (4.4) are coordinates on supersphere S , since the definition of supersphere is satisfied:

‡ 2 xixi + ǫαβθαθβ = (ϕ ϕ) = 1.

Except for the south-pole of S2|2, the Hopf superspinor has an analytic form

1 (1 + x3) 1 θǫθ − 4(1 + x3) 1   ϕ = 1 . 2(1 + x3) (x1 + ix2) 1+ θǫθ   4(1 + x3)    p  (1 + x3)θ1 + (x1 + ix2)θ2      The corresponding connection is derived as

‡ A = iϕ dϕ = dxiAi + dθαAα, − with

1 2+ x3 1 Ai = ǫij xj 1+ θǫθ , Aα = i(xiσiǫθ)α. (4.6) −2(1 + x ) 3 2(1 + x ) −2 3 3

Ai and Aα are the super gauge field of supermonopole. The field strength is evaluated by the formula 1 1 F = dA = dxi dxjFij + dxi dθαFiα dθα dθβFαβ, 2 ∧ ∧ − 2 ∧ where 1 3 Fij = ∂iAj ∂jAi = ǫij xj 1+ θǫθ , − 2 3 2 1 Fiα = ∂iAα ∂αAi = i (θσjǫ)α(δij 3xixj), − − 2 − 3 Fαβ = ∂αAβ + ∂βAα = ixi(σiǫ)αβ 1+ θǫθ . (4.7) − 2 The analytic form of the Hopf superspinor except for the north pole is given by

1 (x ix ) 1+ θǫθ 1 − 2 4(1 x ) 1  − 3  ϕ′ = 1 , 2(1 x3)  (1 x3) 1 θǫθ  −  − − 4(1 x3)   −  p  (x1 ix2)θ1 + (1 x3)θ2   − −    and the corresponding gauge field is obtained as

′ 1 2 x3 ′ 1 A = ǫij xj 1+ − θǫθ , A = i (xiσiǫθ)α = Aα. (4.8) i 2(1 x ) 3 2(1 x ) α − 2 − 3 − 3 Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 13

The field strength F ′ = dA′ is same as (4.7), suggesting the two expressions of the Hopf superspinor are related by the transformation

ϕ′ = ϕ g = g ϕ. · · Here, g denotes (1) gauge group element given by U x ix 1 g = e−iχ = 1 − 2 1+ θǫθ , 1 x2 2(1 x2) − 3 − 3 with χ given by tanp χ = x2 . The (1) group element yields x1 U

∗ 1 1 ig dg = ǫij xidxj 1+ θǫθ , − 1 x2 3 1 x2 − 3 − 3 and A (4.6) and A′ (4.8) are expressed as

1 1 ∗ A = i 1 x (1 + θǫθ) g dg + dθαAα, 2 − 3 2 ′ 1 1 ∗ A = i 1+ x (1 + θǫθ) g dg + dθαAα. − 2 3 2 Then, obviously the gauge fields are related as

A′ = A ig∗dg, − and then

F ′ = F.

4.2 UOSp(1|2) Landau model Next, we consider Landau problem on a supersphere in supermonopole background [39]. The supermonopole gauge field is given by

I 2r + x3 I Ai = ǫij xj 1+ θǫθ , Aα = i(xiσiǫθ)α, −2r(r + x ) 3 2r2(r + x ) −2r3 3 3 I where 2 (I denotes an integer) is a magnetic charge of the supermonopole. Generalizing the SO(3) Landau Hamiltonian (3.8) to UOSp(1 2) form, we have | 1 H = (Λ2 + ǫ Λ Λ ), (4.9) 2MR2 i αβ α β where Λi and Λα are the bosonic and fermionic components of covariant angular momentum: 1 1 1 Λi = iǫijkxjDk + θα(σi)αβDβ, Λα = (ǫσi)αβxiDβ θβ(σi)βαDi, − 2 2 − 2 with Di = ∂i + iAi and Dα = ∂α + iAα. Λi and Λα obey the following graded commutation relations

2 1 2 [Λi, Λj]= iǫijk(Λk r Fk), [Λi, Λα]= (σi)βα(Λβ r Fβ), − 2 − 1 2 Λα, Λβ = (ǫσi)αβ(Λi r Fi), { } 2 − 14 K. Hasebe where Fi and Fα are I I F = x , F = θ . i 2r3 i α 2r3 α Just as in the case of the SO(3) Landau model, the covariant angular momentum is not a conserved quantity, in the sense [Λi,H] = 0, [Λα,H] = 0. Conserved angular momentum is 6 6 constructed as

2 2 Li = Λi + r Fi, Lα = Λα + r Fα.

It is straightforward to see Li and Lα satisfy the UOSp(1 2) algebra (4.3). Since the covariant | angular momentum and the field strength are orthogonal in the supersymmetric sense: ΛiFi + ǫαβΛαFβ = FiΛi + ǫαβFαΛβ = 0, the UOSp(1 2) Landau Hamiltonian (4.9) can be rewritten as | 2 1 2 I H = L + ǫαβLαLβ . 2MR2 i − 4 I With the Casimir index J = 2 + n, the energy eigenvalue is derived as 1 1 I2 1 1 I E = J J + = n n + + I n + , n 2 I 2 2MR 2 − 4 J=n+ 2 2MR 2 4 and the degeneracy in the nth Landau level is

dn = 4J + 1 J n I = 4n + 2I + 1. | = + 2 In particular in the LLL (n = 0), the degeneracy becomes

dLLL = 2I + 1. (4.10) The eigenstates of the UOSp(1 2) Hamiltonian are referred to the supermonopole harmonics, | and the LLL eigenstates are constructed by taking symmetric products of the components of the Hopf superspinor (4.5):

B (m1,m2) I! m1 m2 F (n1,n2) I! n1 n2 ϕLLL = ϕ1 ϕ2 , ϕLLL = ϕ1 ϕ2 η, (4.11) rm1!m2! rn1!n2! where m + m = n + n +1= I (m ,m ,n ,n 0). The total number of ϕB (m1,m2) and 1 2 1 2 1 2 1 2 ≥ LLL ϕF (n1,n2) is (I + 1) + (I) = 2I + 1, which coincides with (4.10). In the LLL, the UOSp(1 2) LLL | angular momentum is reduced to

2 I 2 I Li R Fi = xi, Lα R Fα = θα, → 2R → 2R and coordinates on supersphere are identified with the UOSp(1 2) operators | Xi = αLi, Θα = αLα, which satisfy the algebra defining fuzzy supersphere: α α [Xi, Xj]= iαǫijkXk, [Xi, Θα]= (σi)βαΘβ, Θα, Θβ = (ǫσi)αβXi. (4.12) 2 { } 2 With Lagrange formalism, we reconsider the LLL physics. The present one-particle La- grangian on a supersphere is given by M L = x˙ 2 + ǫ θ˙ θ˙ + A x˙ + θ˙ A 2 i αβ α β i i α α Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 15 with the constraint

2 xixi + ǫαβθαθβ = R . In the LLL, the kinetic term is quenched, and the Lagrangian takes the form of

‡ d L =x ˙ iAi + θ˙αAα = iIϕ ϕ, (4.13) LLL − dt with the constraint

ϕ‡ϕ = 1. (4.14)

From the LLL Lagrangian (4.13), the canonical momentum of ϕ is derived as π = iIϕ∗, and from the commutation relation between ϕ and π, the complex conjugation is quantized as 1 ∂ ϕ∗ = . (4.15) I ∂ϕ After quantization, the normalization condition (4.14) is imposed on the LLL basis: ∂ ϕt ϕ = Iϕ . ∂ϕ LLL LLL One may confirm the LLL basis (4.11) satisfies the condition. By inserting (4.15) to the graded Hopf map (4.4), we obtain ∂ ∂ X = αϕtl , Θ = αϕtl . i i ∂ϕ α α ∂ϕ Apparently, they satisfy the algebra of fuzzy supersphere (4.12). Thus, in the case of the fuzzy two-sphere, the appearance of fuzzy supersphere in LLL is naturally understood in the context of the graded Hopf map.

4.3 Fuzzy supersphere Fuzzy supersphere is constructed by taking symmetric representation of the UOSp(1 2) group | [3, 4, 46, 22]. We first introduce a superspinor extension of the Schwinger operator

ϕˆ1 ϕˆ = ϕˆ ,  2 ηˆ   † whereϕ ˆ1 andϕ ˆ2 are Schwinger boson operators, andη ˆ is a fermion operator: [ϕ ˆi, ϕˆj] = δij, † [ϕ ˆi, ηˆ] = 0 and η,ˆ ηˆ = 1. With such Schwinger superoperator, coordinates on fuzzy super- { } sphere are constructed as

† † Xˆi = αϕˆ liϕ,ˆ Θˆ α = αϕˆ lαϕ.ˆ Square of the radius of fuzzy supersphere is given by α2 Xˆ Xˆ + ǫ Θˆ Θˆ = (ϕ ˆ†ϕˆ)(ϕ ˆ†ϕˆ + 1), i i αβ α β 4 and then the radius is specified by the integer eigenvalue I of the number operator Iˆ ϕˆ†ϕˆ = † † † ≡ ϕˆ1ϕˆ1 +ϕ ˆ2ϕˆ2 +η ˆ ηˆ as α R = I(I + 1). I 2 p 16 K. Hasebe

The UOSp(1 2) symmetric irreducible representation is explicitly constructed as |

1 † m1 † m2 1 † n1 † n2 † m1,m2 = (ϕ ˆ1) (ϕ ˆ2) 0 , n1,n2 = (ϕ ˆ1) (ϕ ˆ2) ηˆ 0 , | i √m1!m2! | i | i √n1!n2! | i with m + m = n + n +1 = I (m ,m ,n ,n 0). Thus, the total number of the states 1 2 1 2 1 2 1 2 ≥ constructing fuzzy supersphere is d = (I + 1)+ (I) = 2I + 1. There is one-to-one correspondence between the supermonopole harmonics in LLL and the states on fuzzy supersphere. Especially, the Hopf superspinor corresponds to the superspin coherent state of Schwinger superoperator.

5 2nd Hopf map and fuzzy four-sphere

In this section, we discuss relations between the 2nd Hopf map

S3 S7 S4 −→ and fuzzy four-sphere. Though the 2nd and 3rd Hopf maps were first introduced by Hopf [29], we follow the realization given by Zhang and Hu [47] in the following discussions.

5.1 2nd Hopf map and SU(2) monopole Realization of the 2nd Hopf map is easily performed by replacing the imaginary unit with the imaginary quaternions, q1, q2, q3. The Pauli matrices (3.1) are promoted to the following quaternionic Pauli matrices 0 q 0 q 0 q γ = − 1 , γ = − 2 , γ = − 3 , 1 q 0 2 q 0 3 q 0 1 2 3 0 1 1 0 γ = , γ = . (5.1) 4 1 0 5 0 1 − γi (i = 1, 2, 3) correspond to σ2, γ4 to σ1, and γ5 to σ3, respectively. With such quaternionic “Pauli matrices”, the 2nd Hopf map is realized as

† ψ ψ γaψ = xa, (5.2) → where a = 1, 2,..., 5, and ψ is a two-component quaternionic (quaternion-valued) spinor satis- † fying the normalization condition ψ ψ = 1. As in the 1st Hopf map, xa given by the map (5.2) † 2 automatically satisfies the condition xaxa = (ψ ψ) = 1. Like the 1st Hopf spinor, the quaternionic Hopf spinor has an analytic form, except for the south pole, as 1 1+ x ψ = 5 , x + q x 2(1 + x5) 4 i i and, exceptp for the north pole, as

1 x qixi ψ′ = 4 − , 2(1 x ) 1 x5 − 5 − where i = 1p, 2, 3 are summed over. These two expressions are related by the transformation

ψ′ = ψ g = g ψ, · · where g is a quaternionic U(1) group element given by

−qiχi χi 1 g = e = cos(χ) qi sin(χ)= (x4 qixi). − χ 1 x2 − − 5 p Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 17

2 χi 1 Here, χi and its magnitude χ = χi are determined by χ tan(χ) = x4 xi. It is possible to pursue the discussions with use ofq quaternions, but for later convenience, we utilize the Pauli matrix representation of the imaginary quaternions:

q = iσ , q = iσ , q = iσ . 1 − 1 2 − 2 3 − 3 The quaternionic U(1) group is usually denoted as U(1, H), and as obvious from the above identification, U(1, H) is isomorphic to SU(2). The quaternionic Pauli matrices (5.1) are now represented by the following SO(5) gamma matrices:

0 iσ 0 iσ 0 iσ γ = 1 , γ = 2 , γ = 3 , 1 iσ 0 2 iσ 0 3 iσ 0 − 1 − 2 − 3 0 1 1 0 γ = , γ = , 4 1 0 5 0 1 − which satisfy γa, γb = 2δab (a, b = 1, 2, 3, 4, 5). Corresponding to the quaternionic Hopf spinor, { } we introduce a SO(5) four-component spinor (the 2nd Hopf spinor)

ψ1 ψ ψ =  2 , (5.3) ψ3 ψ   4   subject to the constraint

ψ†ψ = 1.

With ψ, the 2nd Hopf map (5.2) is rephrased as

† ψ xa = ψ γaψ. (5.4) → 4 It is easy to check that xa satisfies the condition of S :

† 2 xaxa = (ψ ψ) = 1.

An analytic form of the 2nd Hopf spinor, except for the south pole, is given by

1 (1 + x )φ ψ = 5 , (5.5) 2(1 + x ) (x4 iσixi)φ 5 − where φ isp the 1st Hopf spinor representing S3-fibre. The connection is derived as

† † A = iψ dψ = φ dxaAaφ (5.6) − where

1 + Aµ = ηµνixνσi, A5 = 0. (5.7) −2(1 + x5)

+ They are the SU(2) gauge field of Yang monopole [34]. Here, ηµνi signifies the ’tHooft eta- symbol of instanton [48]:

+ η = ǫµνi + δµiδν δµ δνi. µνi 4 4 − 4 18 K. Hasebe

1 The corresponding field strength F = dA + iA A = dxa dxbFab: ∧ 2 ∧ Fab = ∂aAb ∂bAa + i[Aa, Ab], − is evaluated as

1 + Fµν = xµAν + xν Aµ + η σi, Fµ = F µ = (1+ x )Aµ. (5.8) − 2 µνi 5 − 5 5 Another analytic form of the 2nd Hopf spinor, except for the north pole, is given by 1 (x + ix σ )φ ψ′ = 4 i i . 2(1 x ) (1 x5)φ − 5 − The correspondingp connection is calculated as

′ 1 − ′ A = η xνσi, A = 0, (5.9) µ −2(1 x ) µνi 5 − 5 where

− η = ǫµνi δµiδν + δµ δνi, µνi 4 − 4 4 and the field strength is

′ ′ ′ 1 − ′ ′ ′ F = xµA xνA + η σi, F = F = (1 x )A . µν ν − µ 2 µνi µ5 − 5µ − 5 µ As is well known, the ’tHooft eta-symbol satisfies the self (anti-self) dual relation

± 1 ± η = ǫµνρση . µνi ±2 ρσi The two expressions, ψ and ψ′, are related by the SU(2) transition function 1 g = (x4 + ixiσi), √1 x 2 − 5 which yields

† 1 − † 1 + ig dg = η xνσidxµ, idgg = η xνσidxµ, − −1 x 2 µνi − 1 x 2 µνi − 5 − 5 and the gauge fields (5.7) and (5.9) are concisely represented as 1 x 1+ x A = − 5 idgg†, A′ = 5 ig†dg. 2 − 2 Then, A and A′ are related as A′ = g†Ag ig†dg, − and their field strengths are also F ′ = g†F g. This manifests the non-trivial topology of the SU(2) bundle on a four-sphere. In the Language of the homotopy theorem, the non-trivial topology of the SU(2) bundle is expressed by π (SU(2)) Z, 3 ≃ which is specified by the 2nd Chern number 1 c2 = 2 tr (F F ). 2(2π) 4 ∧ ZS Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 19

5.2 SO(5) Landau model We next explore the Landau problem in 5D space [47]. The Landau Hamiltonian is given by

1 1 ∂2 2 ∂ 1 H = D2 = + Λ2 , (5.10) −2N a −2M ∂r2 − Mr ∂r 2Mr2 ab a

Λab = ixaDb + ixbDa. −

As in the previous 3D case, Λab do not satisfy a closed algebra, but satisfy

[Λab, Λcd]= i(δacΛbd + δbdΛac δbcΛad δadΛbc) − − i(xaxcFbd + xbxdFac xbxcFad xaxdFbc), − − − where Fab are given by (5.8). The SO(5) conserved angular momentum is constructed as

2 Lab = Λab + r Fab.

On a four-sphere, the Hamiltonian (5.10) is reduced to the SO(5) Landau Hamiltonian

1 H = Λ2 , 2MR2 ab Xa

I! ψ(m1,m2,m3,m4) = ψm1 ψm2 ψm3 ψm4 (5.13) LLL m !m !m !m ! 1 2 3 4 r 1 2 3 4 3We follow the notation in [49]. 20 K. Hasebe with m + m + m + m = I (m ,m ,m ,m 0). Then, the LLL degeneracy is given by 1 2 3 4 1 2 3 4 ≥ 1 d = (I + 3)(I + 2)(I + 1), LLL 3! which actually coincides with dn=0 of (5.12). Up to now, everything is parallel with the SO(3) Landau model, but emergence of fuzzy four-sphere in LLL is not transparent unlike the fuzzy two-sphere case. With Lagrange formalism, we revisit LLL physics of the SO(5) Landau model. The present one-particle Lagrangian on a four-sphere is given by M L = x˙ x˙ +x ˙ A , (5.14) 2 a a a a with a constraint

2 xaxa = R . (5.15)

In the LLL, the interaction term survives to yield

LLLL =x ˙ aAa, (5.16) and the relation (5.6) implies d L = iψ† ψ, (5.17) LLL − dt and also the constraint (5.15) is rewritten as

ψ†ψ = 1. (5.18)

Interestingly, in the LLL, the original SO(5) symmetry of the Lagrangian (5.14) is enhanced to the SU(4) symmetry: the rotational symmetry of the 2nd Hopf spinor. We treat the 2nd spinor as the fundamental variable and apply the quantization condition. After quantization, ∗ 1 ∂ the complex conjugate spinor is regarded as the derivative ψ = I ∂ψ , and the normalization condition (5.18) is translated to the LLL condition d ψt ψ = Iψ . (5.19) dψ LLL LLL The LLL states (5.13) indeed satisfy the condition (5.19). Here, we comment on the origin of the SU(4) symmetry and its relation to the 2nd Hopf map. In the LLL, the 2nd Hopf spinor plays a primary role, and the total manifold S7 naturally appears in LLL. Projecting out the U(1) phase from S7, we obtain the structure of

CP 3 S7/S1. ≃ This suggests physical equivalence between the LLL of SO(5) Landau model and that of SU(4) Landau model on CP 3. (Detail discussions on physical equivalence between two Lagrangians (5.16) and (5.17) are found in [51], and see Appendix B.2 also.) The appearance of CP 3 can also be understood as follows. As mentioned in Introduction, S4 is not a K¨ahler manifold that accommodates symplectic structure. The “minimally extended” symplectic manifold of S4 is CP 3, which is given by the coset SU(4)/U(3), and then the SU(4) structure naturally appears. Such observation is completely consistent with the mathematical expression of the fuzzy four- sphere, since

S4 SO(5)/U(2) SU(4)/U(3) CP 3, F ≃ ≃ ≃ Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 21 where we used SO(6)/SO(5) S5 U(3)/U(2) and SO(6) SU(4). By inserting the deriva- ≃ ≃ ≃ tive expression of the complex 2nd Hopf spinor to the 2nd Hopf map (5.4), we find that the coordinate on S4 is expressed by the following operator ∂ X = αψtγ . a a ∂ψ As we shall see in the next subsection, the SU(4) structure also appears in the enhanced algebra of Xa.

5.3 Fuzzy four-sphere The fuzzy four-sphere is constructed by taking a fully symmetric representation of the SO(5) spinor [2, 9, 14, 15, 52, 53, 54]. As in the fuzzy two-sphere case, the Schwinger boson formalism is useful to construct coordinates on fuzzy four-sphere α Xˆ = ψˆ†γ ψ,ˆ (5.20) a 2 a ˆ ˆ ˆ† where ψ is a four-component Schwinger boson operator satisfying [ψα, ψβ] = δαβ (α, β = 1, 2, 3, 4). The commutations relation of Xa gives α [Xˆ , Xˆ ]= Xˆ a b 2 ab where Xâb is the SO(5) generator of the form α Xˆ = ψˆ†σ ψˆ (5.21) ab 2 ab i with σab = [γa, γb]. It is important to notice, unlike the case of fuzzy two-sphere, the fuzzy − 4 coordinates do not satisfy a closed algebra by themselves but yield the SO(5) generators. With the SO(5) generators Xâb, the fuzzy coordinates satisfy the following closed algebra, α α [Xâ, Xˆb]= Xâb, [Xâ, Xˆbc]= i (δabXˆc δacXˆb), 2 − 2 − α [Xâb, Xˆcd]= i (δacXˆbd δadXˆbc + δbcXâd δbdXâc). 2 − − ˆ 1 ˆ ˆ ˆ By identifying Xa6 = 2 Xa and Xab = Xab, we find the above algebra is concisely expressed by the SO(6) algebra, α [XÂB, XˆCD]= i (δAC XˆBD δADXˆBC + δBC XÂD δBDXÂC ), 2 − − where A, B = 1, 2,..., 6. Thus, the algebra defining fuzzy four-sphere is SO(6) SU(4). We ≃ encountered the SU(4) structure again, and the fuzzy manifold naturally defined by SU(4) 3 algebra is fuzzy CP (see Appendix B.1). The enhanced SU(4) algebra with extra Xab coordinates accounts for the existence of extra fuzzy-dimensions [14, 15]. Interestingly, CP 3 is locally expressed as the two-sphere fibration over the four-sphere:

CP 3 S4 S2, ≈ × since CP 3 S7/S1 S4 S3/S1. The “extra dimension” of S4 can be understood as two-sphere ≃ ≈ × F fibration over the four-sphere. Square of the radius of fuzzy four-sphere is calculated as α2 Xˆ Xˆ = (ψˆ†ψˆ)(ψˆ†ψˆ + 4), a a 4 22 K. Hasebe and, the radius is α R = I(I + 4), I 2 p ˆ ˆ† ˆ ˆ† ˆ ˆ† ˆ ˆ† ˆ where I signiﬁes an integer eigenvalue of the number operator I ψ ψ = ψ1ψ1 + ψ2ψ2 + ψ3ψ3 + ˆ† ˆ ≡ ψ4ψ4. Similarly, the SO(5) Casimir operator is calculated as

α2 Xˆ 2 = (ψˆ†ψˆ)(ψˆ†ψˆ + 4), ab 8 Xa

6 3rd Hopf map and fuzzy manifolds

Here, we consider realization of the 3rd Hopf map

S7 S15 S8, −→ and corresponding fuzzy manifolds. Unlike the previous cases, there are two kinds of fuzzy manifolds; S8 SO(9)/U(4) and CP 7 SU(8)/U(7), depending on the choice of irreducible F ≃ F ≃ representation of SO(9). The contents in this section are mainly based on Bernevig et al. [55].

6.1 3rd Hopf map and SO(8) monopole The 1st and 2nd Hopf maps were realized by sandwiching Pauli and quaternionic Pauli matrices by spinors. One may expect that such realization can be applied to the 3rd Hopf map. However, it is not so straightforward, since octonions cannot be represented by matrices due to their non- associative property. To begin with, we construct Majorana representation of the SO(9) gamma matrix with the octonion structure constants (Table 1). With e0 = 1, the octonion algebra (2.1) is expressed as

eI eJ = δIJ e + fIJKeK , or eP eQ = fP QReR, − 0 where P, Q, R = 0, 1,..., 7. With use of fP QR, the SO(7) gamma matrices iλI (I = 1, 2,..., 7) − are constructed as

(λI )PQ = fIPQ, − or σ 0 0 0 0 σ 0 0 2 − 3 0 σ2 0 0 σ3 0 0 0 λ1 = i   , λ2 =   , − 0 0 σ2 0 0 0 0 12 −  0 0 0 σ   0 0 1 0   − 2   2      Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 23

0 σ 0 0 0 0 σ 0 − 1 − 3 σ1 0 0 0 0 0 012 λ3 =   , λ4 =   , 0 0 0 iσ σ 0 0 0 − 2 3  0 0 iσ 0   0 1 0 0   2   − 2   0 0 σ 0   0 0 0 1  − 1 − 2 0 0 0 iσ2 0 0 σ3 0 λ5 =   , λ6 =  −  , σ1 0 0 0 0 σ3 0 0  0 iσ 0 0   1 0 0 0   − 2   2   0 0 0 iσ   − 2 0 0 σ1 0 λ7 =  −  . 0 σ1 0 0  iσ 0 0 0   − 2    They are real antisymmetric matrices that satisfy

λI , λJ = 2δIJ . { } − With λ 1 , λ and λI (I = 1, 2,..., 7) are regarded as the SO(8) “Weyl +” gamma matrices. 0 ≡ 8 0 Utilizing λ0 and λI , the SO(9) gamma matrices ΓA are constructed as

ΓI = iλI σ , Γ = 1 σ , Γ = 1 σ , ⊗ 2 8 8 ⊗ 1 9 8 ⊗ 3 or

0 λI 0 18 18 0 ΓI = , Γ8 = , Γ9 = . λI 0 1 0 0 1 − 8 − 8 Again, they are real symmetric matrices that satisfy

ΓA, ΓB = 2δAB , { } where A,B,C = 1, 2,..., 9. The octonion structure constants appear in the off-diagonal elements of ΓI . The SO(9) generators are constructed as 1 ΣAB = i [ΓA, ΓB], (6.1) − 4 or more explicitly

σIJ 0 i λI 0 ΣIJ = , ΣI8 = , 0 σIJ −2 0 λI − i 0 λI 1 0 18 ΣI9 = , Σ89 = i − , 2 λI 0 − 2 18 0 where σIJ are the SO(7) generators 1 σ = i [λ , λ ]. IJ 4 I J

Since ΓA are real matrices, the corresponding SO(9) generators (6.1) are purely imaginary ∗ matrices; Σ = ΣAB. Thus, the present representation is indeed the Majorana representation, AB − in which the charge conjugation matrix is given by unit matrix, and the SO(9) Majorana spinor is simply represented by (16-component) real spinor. The 3rd Hopf spinor is introduced as SO(9) Majorana spinor subject to the normalization condition

ΨtΨ = 1, (6.2) 24 K. Hasebe

15 and the 3rd Hopf spinor is regarded as the coordinate of S . By sandwiching ΓA between the 3rd Hopf spinors, we now realize the 3rd Hopf map as

t Ψ xA =Ψ ΓAΨ. (6.3) → 8 xA in (6.3) are coordinates on S , since

t 2 xAxA = (Ψ Ψ) = 1. A , ,..., =1X2 9 An analytic form of Ψ, except for the south pole, is represented as

1 (1 + x )Φ Ψ= 9 , 2(1 + x ) (x8 λI xI )Φ 9 − where Φ isp a SO(7) real 8-component spinor subject to the constraint

ΦtΦ = 1, representing the S7-fibre. Then, Φ has the same degrees of freedom of the 2nd Hopf spinor ψ. We may assign Φ = (Re ψ, Im ψ)t, and the 3rd Hopf spinor is expressed as

Re ψ (1 + x ) 1 9 Im ψ Ψ=   . 2(1 + x9) Re ψ (x λI xI )  8 − Im ψ    p   Naively anticipated connection A = iΨtdΨ vanishes due to the Majorana property of Ψ, − however, defining

1 (1 + x )1 Ψ = 9 8 , (6.4) 2(1 + x ) x818 λI xI 9 − the connectionp of S7-fibre is evaluated as

t A = iΨ dΨ = dxAAA, − where AA = (AM , A ) (M = 1, 2, , 8) are 9 · · · 1 AM = σMN xN , A9 = 0, (6.5) −1+ x9 with M = 1, 2,..., 8. Here, σMN are SO(8) “Weyl +” generators given by 1 1 σIJ = i [λI , λJ ], σI = σ I = i λI . 4 8 − 8 − 2

(σIJ and σI are pure imaginary antisymmetric matrices.) The field strength FAB = ∂AAB 8 − ∂BAA + i[AA, AB] is also evaluated as

FMN = xM AN + xN AM + σMN , FM = F M = (1+ x )AM , − 9 − 9 9 which represent the SO(8) monopole gauge field [35]. Similarly, except for the north pole, the 3rd Hopf spinor has an analytic form

1 x 1 + λ x Ψ′ = 8 8 I I , (6.6) 2(1 x ) (1 x9)18 − 9 − p Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 25 and the connection is

′ 1 A = σ¯MN xN , A = 0, (6.7) M −1 x 9 − 9 where

σ¯IJ = σIJ , σ¯I = σ¯ I = σI . 8 − 8 8 The corresponding field strength is derived as

′ ′ ′ ′ ′ ′ F = xM A xN A +σ ¯MN , F = F = (1 x )A . MN N − M M9 − 9M − 9 M

Here, the SO(8) generators σMN (¯σMN ) satisfy a generalized self (anti-self) dual relation:

4 4 σMN = ǫMNPQABCDσPQσABσCD, σ¯MN = ǫMNPQABCDσ¯PQσ¯ABσ¯CD. 6! −6! The two expressions (6.4) and (6.6) are related by

g 0 Ψ′ = Ψ, 0 g · where g signifies an SO(8) group element

1 g = (x8 + λI xI ), 1 x2 − 9 which yieldsp

t 2 − t 2 + ig dg = Σ xN dxM , igdg = Σ xN dxM . − −1 x 2 MN − 1 x 2 MN − 9 − 9 Then, the gauge fields, (6.5) and (6.7), are concisely represented as

1 1 A = i (1 x )dggt, A′ = i (1 + x )gtdg, 2 − 9 − 2 9 and are related by

A′ = gtAg igtdg. − Similarly, their field strengths are

F ′ = gtF g.

The non-trivial topological structure of the SO(8) monopole bundle is guaranteed by the homotopy theorem

π (SO(8)) Z Z, 7 ≃ ⊕ which is specified by the Euler number

e = tr (F F F F ). 8 ∧ ∧ ∧ ZS 26 K. Hasebe

Table 2. In the 1st and 2nd Hopf maps, the fuzzy manifolds are uniquely determined, since the cor- 2 1 4 3 responding fuzzy spheres and complex projective spaces are equivalent, i.e. SF CPF and SF CPF . 8 ≃ ≃ 7 Meanwhile, in the 3rd Hopf map, two corresponding fuzzy manifolds, SF SO(9)/U(4) and CPF ≃ ≃ S15/S1, are not equivalent. Division algebra Complex numbers Quaternions Octonions Hopf maps 1st 2nd 3rd Fuzzy sphere S2 SO(3)/U(1) S4 SO(5)/U(2) S8 SO(9)/U(4) F ≃ F ≃ F ≃ Fuzzy CP n CP 1 S3/S1 CP 3 S7/S1 CP 7 S15/S1 F ≃ F ≃ F ≃

6.2 Fuzzy CP 7 and fuzzy S8 In the realization of the 3rd Hopf map, we utilized the real (Majorana) spinor. The fuzzification procedure in the previous sections can not be straightforwardly applied to the present case, since we do not have the complex conjugate spinor to be identified with derivative. However, with 16 real components of the 3rd Hopf spinor, we can construct an 8-component normalized complex spinor to be identified with coordinates on CP 7. Then, in the present case, there exist two different types of fuzzy manifolds, depending on the choice of the irreducible representation of SO(9). The first one is the above mentioned fuzzy CP 7 S15/S1 specified by the vector ≃ representation of SO(9), while the other is the fuzzy eight-sphere S8 SO(9)/U(4) specified F ≃ by the spinor representation. Both two fuzzy manifolds are reasonable generalizations of the previous low dimensional fuzzy spheres and fuzzy complex projective spaces (see Table 2).

7 Beyond Hopf maps: even higher dimensional generalization

We have reviewed the construction of fuzzy manifolds based on the Hopf maps. Since the Hopf maps are only three kinds, the corresponding fuzzy manifolds are also limited. However, from the results of the Hopf maps, one may naturally infer two possible generalizations of the fuzzy manifolds, one of which is a series of fuzzy spheres:

S2k SO(2k + 1)/U(k), F ≃ and the other is that of fuzzy complex projective spaces:

CP k SU(k + 1)/U(k) S2k+1/S1. F ≃ ≃ 2k In this section, we first introduce mathematics of fuzzy spheres SF in arbitrary even dimensions, and next provide their physical interpretations, mainly based on Hasebe and Kimura [56] (see also Fabinger [57] and Meng [58]). Fuzzy CP k manifolds and their corresponding Landau models are discussed in Appendix B.

7.1 Clifford algebra: another generalization of complex numbers It may be worthwhile to begin with the story of generalization of the complex numbers to Clifford algebra. Generalization from fuzzy two-sphere to its higher dimensional cousins are quite analogous to the generalization of complex numbers. As discussed in Section 2, Cayley– Dickson construction provides one systematic way to construct new numbers by duplicating the original numbers, but this construction method has a fatal problem: If we utilize the method, the resulting algebra loses a nice property of numbers one by one. For instance, in the construction of quaternions, the commutativity of the complex numbers was lost. In the construction of octonions, even the associativity of quaternions was abandoned. Consequently, generalization Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 27 of complex numbers ends up with the octonions, and there are only three division algebras (except for real numbers). Clifford found another generalization of complex numbers, based on Hamilton’s quaternions and Grassmann algebras, known as Clifford algebra. The Clifford n algebra, Cliffn, consists of 2 basis elements 1, ea, eaeb, eaebeb,...,e e e en (a = b, a = b = { 1 2 3 · · · } 6 6 6 c, . . . ), and its algebraic structure is determined by the relation4

ea, eb = 2δab. (7.1) { } The division algebras except for the octonions are realized as special cases of Clifford algebra, i.e. R = Cliff0, C = Cliff1, and H = Cliff2. The Clifford algebra can also be regarded as the “quantized” Grassmann algebra (compare (7.1) with ηa, ηb = 0 (4.1)). Though the division { } property is in general lost, the Clifford algebras always maintain the nice associative property, and are represented by gamma matrices that satisfy

γa, γb = 2δab. { } Importantly, there are analogous geometrical properties between the division algebras and the Clifford algebras: In the division algebras, new numbers are constructed by the Cayley–Dickson construction. Similarly, higher dimensional gamma matrices are constructed by the lower dimen- (2k−1) sional gamma matrices. Specifically, SO(2k 1) gamma matrices γi (i = 1, 2,..., 2k 1) −(2k+1) − are provided, SO(2k + 1) gamma matrices γa (a = 1, 2,..., 2k + 1) can be constructed as

(2k−1) (2k+1) 0 iγi (2k+1) 0 1 (2k+1) 1 0 γi = (2k−1) , γ2k = , γ2k+1 = . (7.2) iγ 0 ! 1 0 0 1 − i − Thus, in any higher dimensional gamma matrices are constructed by repeating the above procedure from the SO(3) gamma matrices, γ(3) = σ , γ(3) = σ , γ(3) = σ . As the Hopf maps 1 − 2 2 1 3 3 exhibit the hierarchical structure stemming from the Cayley–Dickson construction, the geometry of higher dimensional fuzzy spheres reflects the iterative construction structure of the gamma matrices as we shall see below.

7.2 Mathematical aspects of fuzzy sphere We first introduce mathematical construction of fuzzy spheres [2, 9, 14, 15, 52, 53, 54]. Co- ordinates on fuzzy 2k-sphere are constructed by the SO(2k + 1) gamma matrices in the fully I I I symmetric representation [ 2 , 2 ,..., 2 ]: α Xa = (γa 1 1+ γa 1 1+ + 1 1 γa) , 2 ⊗ ⊗···⊗ ⊗ ⊗···⊗ · · · ⊗···⊗ ⊗ sym where γa (a = 1, 2,..., 2k + 1) are the SO(2k + 1) gamma matrices in the fundamental representation, and the number of the tensor product is I. Square of the radius of fuzzy 2k-sphere is given by

2k+1 α2 X X = I(I + 2k). a a 4 a X=1 Similarly, in the symmetric representation, the eigenvalue of the SO(2k + 1) Casimir is

2k+1 α2 X2 = kI(I + 2k), ab 16 a

1 where Xab = i [Xa, Xb] are the SO(2k +1) generators. (Detail calculation techniques for sym- − 4 metric representation can be found in [21].) Thus, the index I of the symmetric representation determines the magnitude of the radius of fuzzy sphere. As the dimension of the symmetric representation becomes “larger”, the corresponding fuzzy sphere becomes “larger”. As in the case of fuzzy-four sphere, the fuzzy coordinates Xa do not satisfy a closed algebra by themselves, but Xa and Xab satisfy the following enlarged algebra α [Xa, Xb] = 2iαXab, [Xa, Xbc]= i (δabXc δacXb), − 2 − α [Xab, Xcd]= i (δacXbd δadXbc + δbcXad δbdXac). 2 − −

1 With identification Xa, k = X k ,a = Xa and Xab = Xab, the above algebra is found to 2 +2 − 2 +2 2 be equivalent to the SO(2k + 2) algebra α [XAB, XCD]= i (δAC XBD δADXBC + δBC XAD δBDXAC ), 2 − −

2k+1 where A,B,C,D = 1, 2,..., 2k + 2. Thus, the algebra of fuzzy sphere SF is SO(2k + 2) [14], 2k and SF is expressed as

S2k SO(2k + 2)/U(k + 1) SO(2k + 1)/U(k), F ≃ ≃ where we used the relation SO(2k + 2)/SO(2k + 1) S2k+1 U(k + 1)/U(k). The above coset ≃ ≃ representation can also be expressed as

S2k SO(2k + 1)/SO(2k) SO(2k 1)/U(k 1) S2k S2k−2. (7.3) F ≈ × − − ≈ × F

Fuzzy 2k-sphere is constructed not only by the operators Xa but also the “extra” operators Xab, 2k−2 2k and the very existence of Xab brings the extra fuzzy-space SF over S . From (7.3), one may find the fuzzy sphere is expressed by the hierarchical fibrations of lower dimensional spheres

S2k S2k S2k−2 S4 S2, (7.4) F ≈ × ×···× × which reflects the iterative construction of gamma matrices from lower dimensions.

7.3 Hopf spinor matrix and SO(2k) monopole To obtain monopole bundles in generic even dimensions, we “extend” the Hopf maps. First, we define “Hopf spinor matrix” of the form

1 (1 + x )1 Ψ = 2k+1 , (7.5) 2(1 + x ) (x2k1 iγixi) 2k+1 − p k k where 1 stands for 2 2 unit matrix, γi (i = 1, 2,..., 2k 1) are SO(2k 1) gamma matrices, × − 2k+1− and xa (a = 1, 2,..., 2k + 1) are coordinate on 2k-sphere satisfying a=1 xaxa = 1. The Hopf spinor matrix is a 2k+1 2k matrix that satisfies × P † xa1 = Ψ γaΨ, where γa are SO(2k + 1) gamma matrices (7.2). The corresponding monopole gauge field is evaluated as

† A = dxaAa = iΨ dΨ, − Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 29 where

1 + Aµ = Σµνxν, A2k+1 = 0. (7.6) −2(1 + x2k+1)

+ Σµν (µ,ν = 1, 2,..., 2k) are SO(2k) generators given by

+ 1 + + 1 Σ = i [γi, γj], Σ = Σ = γi. ij − 4 i2k − 2ki 2

The field strength Fab = ∂aAb ∂bAa + i[Aa, Ab] is calculated to yield − + Fµν = xµAν + xν Aµ +Σ , Fµ, k = F k ,µ = (1+ x )Aµ. − µν 2 +1 − 2 +1 9 These are the SO(2k) non-Abelian monopole gauge field strength [59, 60, 61]. Another representation of the Hopf spinor matrix is introduced as

1 (x 1 + iγ x ) Ψ′ = 2k i i , (7.7) 2(1 x k ) (1 x2k+1)1 − 2 +1 − and the correspondingp gauge field, A′ = iΨ†dΨ′, reads as − ′ 1 − ′ Aµ = Σµνxν, A2k+1 = 0, (7.8) −2(1 x k ) − 2 +1 where

Σ− =Σ+, Σ− = Σ+ . ij ij i,2k − 2k,i ′ The gauge field strength Fab is

′ ′ ′ − ′ ′ ′ F = xµA xνA +Σ , F = F = (1 x )A . µν ν − µ µν µ,2k+1 − 2k+1,µ − 9 µ + − The SO(2k) generators, Σµν and Σµν, satisfy the generalized self and anti-self dual relations, respectively:

k−1 ± 2 ± ± Σ = ǫµ µ µ µ µ − µ Σ Σ . µ1µ2 ±(2k 2)! 1 2 3 4··· 2k 1 2k µ3µ4 · · · µ2k−1µ2k − The two Hopf spinor matrices (7.5) and (7.7) are related by the transformation

g 0 Ψ′ = Ψ, 0 g · where g is an SO(2k) group element 1 g = (x2k + iγixi). 1 x2 − 2k+1 q With g, the gauge fields (7.6) and (7.8) are concisely represented as

1 † ′ 1 † A = i (1 x k )dgg , A = i (1 + x k )g dg, 2 − 2 +1 − 2 2 +1 where

† 2 − † 2 + ig dg = Σ xνdxµ, igdg = Σ xνdxµ. − −1 x2 µν − 1 x2 µν − 2k+1 − 2k+1 30 K. Hasebe

Then, the two gauge fields are related as

A′ = g†Ag ig†dg, − and the field strengths are

F ′ = g†F g.

The homotopy theorem guarantees the non-trivial topology of the SO(2k) bundle ﬁbration over S2k:

π2k−1(SO(2k)) = Z, which is specified by the Euler number

e = tr F k . 2k ZS 7.4 SO(2k + 1) Landau model In generic d-dimensional space, Landau Hamiltonian is given by [56]

1 1 ∂2 1 ∂ 1 H = D2 = (d 1) + Λ2 , (7.9) −2M a −2M ∂r2 − − 2Mr ∂r 2Mr2 ab Xa

[Λab, Λcd]= i(δacΛbd + δbdΛac δbcΛad δadΛbc) − − i(xaxcFbd + xbxdFac xbxcFad xaxdFbc). − − − The conserved SO(2k + 1) angular momentum is constructed as

2 Lab = Λab + r Fab, which satisfies the SO(2k + 1) algebra

[Lab,Lcd]= i(δacLbd + δbdLac δbcLad δadLbc), − − and generates the SO(2k + 1) transformations, for instance

[Lab, Λcd]= i(δacΛbd + δbdΛac δbcΛad δadΛbc), − − [Lab, Fcd]= i(δacFbd + δbdFac δbcFad δadFbc). − − On 2k-sphere, the Landau Hamiltonian (7.9) is reduced to SO(2k + 1) Landau Hamiltonian

1 H = Λ2 . 2MR2 ab a

With the orthogonality ΛabFab = FabΛab, the Hamiltonian is rewritten as

1 1 H = L2 F 2 = L2 Σ2 , 2MR2 ab ab 2MR2 ab µν − ! − µ<ν ! Xa

Table 3. The fuzzy 2k-sphere is physically realized in the LLL of the SO(2k + 1) Landau model. Previously encountered monopoles are understood as the special cases of SO(2k) monopoles, for instance U(1) SO(2), SU(2)( SU(2)) SO(4). Note also SU(4) SO(6). The holonomy groups of spheres ≃ ⊗ ≃ ≃ are equal to the corresponding monopole gauge groups. Fuzzy sphere Corresponding original sphere Monopole gauge group 2 2 SF SO(3)/U(1) S SO(3)/SO(2) U(1) 4 ≃ 4 ≃ SF SO(5)/U(2) S SO(5)/SO(4) SU(2) 6 ≃ 6 ≃ SF SO(7)/U(3) S SO(7)/SO(6) SU(4) 8 ≃ 8 ≃ SF SO(9)/U(4) S SO(9)/SO(8) SO(8) ≃ . ≃ . . . . . S2k SO(2k + 1)/U(k) S2k SO(2k + 1)/SO(2k) SO(2k) F ≃ ≃

2 2 where a

2 k C k (n,I/2) = n + n(I + 2k 1) + I(I + 2k), 2 +1 − 4 and C2k(I/2) is the SO(2k) Casimir eigenvalue for (I/2): 1 C (I/2) = Σ2 = Ik(I + 2k 2). 2k µν 4 µ<ν − X Consequently, the energy eigenvalues of the SO(2k + 1) Landau model are derived as

1 2 1 En = (n + n(I + 2k 1) + Ik). 2MR2 − 2 In the thermodynamic limit: R,I with I/R2 fixed, the Landau levels are reduced to →∞ I 1 En (n + k). → 2MR2 2 I The LLL energy, ELLL = 4MR2 k, depends on the spacial dimension 2k. In the thermodynamic 2k B I limit, S is reduced to 2k-dimensional plane, and the zero-point energy 2M = 4MR2 coming from each 2-dimensional plane amounts to ELLL. As discussed above, coordinates on fuzzy 2k-sphere are given by the SO(2k + 1) gamma matrices in the symmetric spinor representation. Similarly, the LLL basis of the SO(2k + 1) Landau model realizes such a symmetric spinor representation. Thus, the LLL of SO(2k + 1) Landau model provides a physical set-up for 2k-dimensional fuzzy sphere (see Table 3).

7.5 Dimensional hierarchy Here, we give a physical interpretation of the hierarchical geometry of higher dimensional fuzzy spheres (7.4). From the formula of the irreducible representation of SO(2k + 1) [49], the degeneracy in nth LL is given by k−1 k k−2 k 2n+I +2k 1 (n+k 1)! n+I +2k i I +2k i l d(n)= − − (I +2i 1) − − − . (2k 1)!! n!(k 1)! − 2k i Ik i l i i − − Y=1 Y=2 − Yl=1 i=Yl+2 − − 32 K. Hasebe

Figure 1. Lower dimensional spheres gather spherically to form a higher dimensional fuzzy sphere.

Table 4. Correspondence between algebras, monopoles, and fuzzy spheres. Algebras Bundle structure Monopoles Fuzzy manifolds Division algebra Canonical bundle U(1), SU(2), SO(8) Fuzzy 2,4,8-spheres Grassmann algebra Graded canonical bundle Supermonopole Fuzzy supersphere Clifford algebra Spinor bundle SO(2k) Higher d. fuzzy spheres

I I I In the LLL, the representation is reduced to the fully symmetric spinor representation [ 2 , 2 ,. . ., 2 ], and the degeneracy becomes

k−1 (I + 2k 1)!! (I + 2l)!l! k k 1 k k d = − I I2 I3 I −1 I = I 2 ( +1). (7.10) LLL (2k 1)!!(I 1)!! (I + l)!(2l)! −→ · · · · · · l − − Y=1 The last expression implies a nice intuitive picture of the hierarchical geometry of fuzzy spheres. 2k 2k k 2 k Each of the SO(2k) monopole fluxes on S occupies an area ℓB = (1/B) = (2R /I) with the magnetic field B = (2πI)/(4πR2), and the number of fluxes on S2k is R2k/ℓ2k Ik. Besides, ∼ B ∼ the monopole flux itself is represented by the generators of the non-Abelian SO(2k) group, and is regarded as a (2k 2)-dimensional fuzzy sphere. Again, the (2k 2)-dimensional fuzzy sphere − − is interpreted as a (2k 2)-dimensional sphere in SO(2k 2) monopole background, and then, 2k−2 − − 2k−2 k−1 on S , there are SO(2k 2) fluxes each of which occupies the area ℓB = (1/B) = k − (2R2/I) −1. Therefore, the number of SO(2k 2) fluxes on S2k−2 is R2k−2/ℓ2k−2 Ik−1. − ∼ B ∼ Similarly, the SO(2k 2) non-Abelian flux is given by the generators of the SO(2k 2) group, − − and regarded as a fuzzy S2k−4. Thus, on S2k, we have Ik S2k−2, on which Ik−1 S2k−4, on which Ik−2 S2k−6, . . . . By this iteration, we obtain the formula (7.10). Inversely, we can view this mechanism from low dimensions: Lower dimensional spheres gather spherically to form a higher dimensional sphere, and such iterative process amounts to construct a higher dimensional fuzzy sphere. The dimensional hierarchy is depicted in Fig. 1.

8 Summary and discussions

We reviewed the close relations between monopoles, LLL, and fuzzy spheres. The fuzzy 2k-sphere is physically realized in the LLL of the SO(2k + 1) Landau model. In the generalization of fuzzy spheres, three classical algebras; division algebra, Grassmann algebra, and Clifford algebra, played crucial roles. They brought the basic structures of monopole bundles and fuzzy spheres (Table 4). In particular, the hierarchical geometry of fuzzy spheres is a direct manifestation Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 33 of their gamma matrix construction: As higher dimensional gamma matrices are constructed by lower dimensional gamma matrices, higher dimensional fuzzy sphere is constructed by lower dimensional spheres. Such dimensional hierarchy can be physically understood in the context of higher dimensional Landau model. It should be mentioned that, in [62], such interpretation was successfully applied to dual description of higher dimensional fuzzy spheres in string theory. We also emphasize the importance of the Hopf map in realizing fuzzy sphere. In the fuzzification of spheres, the total manifolds (Hopf spinor spaces) of the Hopf maps played a fundamental role: The total manifolds were firstly fuzzificated and as “a consequence” the basemanifolds (spheres) are fuzzificated. Interestingly, this fuzzification mechanism coincides with the philosophy of twistor theory (see [63] and references therein). There are various works related to the present paper: Even restricted to the recent ones, supersymmetric extensions of the Landau model [64, 65, 66, 67, 68, 69], supersymmetric fuzzy manifolds [70, 71], generalizations of the Hopf maps [72, 73, 74, 75], supersymmetric quantum mechanics [76, 77, 78, 79, 80, 81, 82, 83], and applications to string theory [84, 85, 86, 87]. Finally, we comment on applications to many-body physics. The correspondence between fuzzy geometry and LLL physics argued in the paper was at one-particle level observation. Interestingly, there even exists correspondence at many-body level: Many-body groundstate wavefunction of quantum Hall effect (Laughlin wavefunction) is mathematically analogous to an antiferromagnetic ground state (AKLT state) [88, 89]. Accompanied with the higher dimensional and supersymmetric generalizations of the quantum Hall effect5, their formalism has begun to be applied to the construction of antiferromagnetic quantum spin states with higher symmetries [89, 94].

A 0th Hopf map and SO(2) “Landau model”

Real numbers are the “0th” member of the division algebra. For completeness, we introduce the 0th Hopf map

Z S1 2 S1, −→ and the corresponding Landau model. The 0th Hopf map is realized by identifying “opposite points” on a circle, and is simply visualized as the geometry of M¨obius strip whose baseman- 1 ifold is S and transition function is Z2. Unlike the other Hopf maps, the dimension of the basemanifold is odd and the structure group is a discrete group.

A.1 Realization of the 0th Hopf map t t With the coordinate on a circle, w = (w1, w2) (a real two-component spinor subject to w w = 2 2 w1 + w2 = 1), the 0th Hopf map is realized as w w = 1 x = wtσ w, x = wtσ w. (A.1) w → 1 3 2 1 2 Here, σ1 and σ3 are the real Pauli matrices. x1 and x2 are invariant under the transformation (w1, w2) (w1, w2), and hence (x1,x2) is Z2 projection of (w1, w2). From (A.1), one may find 2 2 →−t 2 x1 + x2 = (w w) = 1. Inverting the map, w can be expressed as θ 1 1+ x cos w = 1 = 2 , x  θ  2(1 + x1) 2 sin  2  p   5Interested readers are expected to consult the review [24], and [90, 91, 92, 93, 63] for more recent works. 34 K. Hasebe where x1 and x2 are parameterized as x1 = cos θ and y = sin θ. The corresponding connection i θ vanishes: A = iwtdw = 0. Meanwhile, using a U(1) element ω = w + iw = e 2 , the 0th Hopf − 1 2 map is restated as

ω x + ix = ω2, → 1 2 ∗ t and the connection, A = iω dω = iw σ dw = dxAx + dyAy, is given by − 2 1 1 1 Ax = y, Ay = x or Aθ = . −2 2 −2

It is straightforward to see the field strength, B = ∂xAy ∂yAx, represents a solenoid-like − magnetic field at the origin:

B = πδ(x,y).

A.2 SO(2) “Landau model” Next, we introduce “Landau model” on a circle in the presence of magnetic fluxes [95]

B = Iπδ(x,y).

Here, I is the number of magnetic fluxes, and takes an integer value. The corresponding gauge field is given by I I Ax = y, Ay = x, −2r2 2r2 where r2 = x2 +y2. Since the magnetic fluxes are at the origin, the classical motion of a charged particle on a circle is not affected by the existence of the magnetic fluxes. In quantum mechanics, however, the result is different. With the covariant derivatives Dx = ∂x +iAx and Dy = ∂y +iAy, Landau Hamiltonian on 2D plane is given by 1 1 ∂2 1 H = D2 + D2 = + Λ2, −2M x y −2M ∂r2 2Mr2 where Λ is the covariant angular momentum ∂ Λ= ixDy + iyDx = i + Aθ, − − ∂θ I with Aθ = 2 . On a circle with radius R, the Hamiltonian is reduced to SO(2) “Landau Hamil- tonian” 1 1 ∂ i 2 H = Λ2 = + I . 2MR2 −2MR2 ∂θ 2 This is a one-dimensional quantum mechanical Hamiltonian easily solved. In higher dimensions, the covariant angular momentum is not a conserved quantity, but the present case it is, as simply verified [Λ,H] = 0. Since the magnetic field angular momentum does not exist on the circle, the particle angular momentum itself is conserved. Imposing the periodic boundary condition

u(θ = 2π)= u(θ = 0), the eigenvalue problem is classified to two cases: even I and odd I. For even I, the energy eigenvalue are given by 1 E = n2, n 2MR2 Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 35 where n = 0, 1, 2, 3,... . The eigenstates are

1 i ±n− I θ u(θ)±n = e 2 . √2π Except for the lowest energy level n = 0, every excited energy level is two-fold degenerate. The physical origin of the double degeneracy comes from the left and the right movers on the circle. The energy levels are identical to those of free particle on a circle. Thus, for even I, the magnetic flux does not affect the energy spectrum of the system. Meanwhile, for odd I, the energy eigenvalues become

1 1 2 E′ = n + , (A.2) n 2MR2 2 and the corresponding eigenstates are

′ 1 i ± n+ 1 − I θ u (θ)±n = e 2 2 . √2π All the Landau levels are doubly degenerate even for n = 0. The energy spectra (A.2) are different from those of the free particle on a circle, reflecting the particular role of the gauge field in quantum mechanics. It is also noted that the number of the magnetic fluxes I has nothing to do with the degeneracy in Landau levels unlike the other Landau models (see equation (3.11) for instance).

B Generalized 1st Hopf map and SU(k + 1) Landau model

B.1 SU(k + 1) Landau model and fuzzy complex projective space The projection from sphere to complex projective space,

S1 S2k+1 CP k, −→ is a straightforward generalization of the 1st Hopf map. (Note CP 1 S2.) In [96], Karabali ≃ and Nair introduced SU(k +1) Landau model on CP k in U(1) monopole background. Since the SU(k + 1) Landau models are reviewed in [20, 24], we survey the main results. The SU(k + 1) Landau model Hamiltonian is given by

1 k H = C I2 , (B.1) 2MR2 SU(k+1) − 2(k + 1) where CSU(k+1) represents the Casimir operator of the SU(k + 1) group. One may notice analogies between (B.1) and the spherical Landau model Hamiltonians (see for instance (3.9)). The SU(k +1) group has k Casimirs, and hence its irreducible representations are specified with Young tableau index, [λ1, λ2, . . . , λk]. In the SU(k + 1) Landau model, Young tableau index is chosen as [λ1, λ2, . . . , λk] = [p + q,q,...,q]. When q = 0, the Young tableau index becomes [p, 0,..., 0] representing the fully symmetric representation of the SU(k + 1) spinor. Meanwhile, when p = 0, the index becomes [q,q,...,q] representing the fully symmetric representation of the SU(k + 1) complex spinor. The “difference” between the fundamental and complex representations is specified by I = p q, which corresponds to the U(1) monopole charge. The − Landau level index can be taken as q = n, since, in LLL (n = 0), the irreducible representation is reduced to the fully symmetric spinor representation. Thus, the SU(k + 1) Young tableau 36 K. Hasebe index (n,I) [2n + I,n,...,n] specifies the states of the nth Landau level in U(1) monopole ≡ k background with magnetic charge 2(k+1) I. Then, the energy eigenvalues are derived as q 1 k 2 1 k En = C (n,I) I = n(n + k)+ I n + (B.2) 2MR2 SU(k+1) − 2(k + 1) 2MR2 2 and the corresponding degeneracies are

1 (I + n + k 1)!(n + k 1)! d(n,I)= (2n + I + k) − − . (B.3) k!(k 1)! (I + n)!n! − I The k dependence of the LLL energy ELLL = 4MR2 k is simply understood by remembering that CP k consists of k complex planes each of which equally provides zero-point energy contribu- B I tion 2M = 4MR2 to LLL energy. The LLL basis elements are constructed by the fully symmetric products of the SU(k + 1) Hopf spinor or the Perelomov coherent state of SU(k + 1) [97]

1 z1 1   z2 ψ = ∗ ∗ ∗ , (B.4) 1+ z z1 + z z2 + + z zk  .  1 2 · · · k  .    p z   k   as

(m1,m2,...,mk) 1 m1 m2 m3 mk+1 ψ = 1 z1 z2 zk , (B.5) LLL ∗ ∗ ∗ I 1+ z z + z z + + z zk · · · 1 1 2 2 · · · k p where m1 + m2 + + mk+1 = I (m1,m2,...,mk 0). Thus, the degeneracy in the LLL is (I+k)! · · · ≥ dLLL = I!k! that coincides with d(0,I) of (B.3). With the SU(k + 1) Hopf spinor (B.4), the LLL Lagrangian is given by

d L = iIψ† ψ. LLL − dt

After quantization, the normalization constraint ψ†ψ = 1 is imposed on the LLL states:

d ψt ψ = Iψ . dψ LLL LLL

Indeed, the LLL states (B.5) satisfy the above condition. Fuzzy CP k is constructed by taking the fully symmetric representation of SU(k + 1) [98, 99, 100, 22]. With the SU(k + 1) extended Schwinger boson operator

ψˆ1 ψˆ2 ψˆ =  .  .   ψˆ   k+1   k satisfying [ψˆα, ψˆβ]= δαβ (α, β = 1, 2,...,k + 1), coordinates on fuzzy CP are constructed as

† Tˆi = αψˆ tiψˆ Hopf Maps, Lowest Landau Level, and Fuzzy Spheres 37

2 where ti (i = 1, 2,...,k +2k) stand for the fundamental representation matrix of the SU(k +1) 1 C k generators, normalized as tr (titj) = 2 δij. Square of the radius of fuzzy P is given by the SU(k + 1) Casimir operator:

k2+2k k Tˆ Tˆ = α2I(I + k + 1). (B.6) i i 2(k + 1) i X=1 In particular for fuzzy CP 1 (k = 1), equation (B.6) reproduces the result of fuzzy two-sphere (3.19):

3 α2 Tˆ Tˆ = I(I + 2). i i 4 i X=1 Similarly for fuzzy CP 3 (k = 3), equation (B.6) becomes

15 3 Tˆ Tˆ = α2I(I + 4), i i 8 i X=1 which is equal to the Casimir of fuzzy four-sphere

5 1 3 Xˆ Xˆ + Xˆ Xˆ = α2I(I + 4), 4 a a ab ab 8 a=1 X Xa

B.2 Relations to spherical Landau models As easily verified, the SU(2) Landau model is equivalent to the SO(3) Landau model. From the formulae (B.2) and (B.3), the energy eigenvalues and degeneracies of the SU(2) Landau Hamiltonian are derived as 1 1 E = n(n +1)+ I n + , d = 2n + I + 1. n 2MR2 2 n These indeed reproduce the results of the SO(3) Landau model, (3.10) and (3.11). Moreover, the SU(2) coherent state

1 1 ψ = , √1+ z∗z z corresponds to the 1st Hopf spinor (3.4) by the relation φ x + ix z = 2 = 1 2 , φ1 1+ x3 and the SU(2) LLL basis elements (B.5) for k = 1 are equal to equation (3.12). 38 K. Hasebe

There also exists correspondence between the SU(4) and SO(5) Landau models in the LLL. Again from equations (B.2) and (B.3), the energy eigenvalues and degeneracies of the SU(4) Landau model are read as 1 3 E = n(n +3)+ I n + , n 2MR2 2 1 d = (n + 1)(n + 2)(I + n + 1)(I + n + 2)(I + 2n + 3). (B.7) n 12 The SU(4) Landau level energy is different from that of the SO(5) Landau model (5.11), only 1 by the total energy shift 4MR2 I. This discrepancy is understood by a simple geometrical argu- ment [96]: Since CP 3 S4 S2, the CP 3 is regarded as S2-fibration over S4, and the zero-point ≈2 × B 1 energy from the extra S -space, 2M = 4MR2 I, gives rise to the difference. Generally, the degeneracy of Landau levels of the SU(4) Landau model (B.7) is different from that of the SO(5) Lan- 1 dau model (5.12), but in the LLL, both quantities coincide to yield dLLL = 6 (I +1)(I +2)(I +3). This manifests the equivalence between the SO(5) and SU(4) Landau models in LLL. By com- paring the 2nd Hopf spinor (5.5) with the SU(4) coherent state

1 1 z ψ =  1 , ∗ ∗ ∗ z 1+ z1z1 + z2 z2 + z3z3 2 z   3 p   we find the correspondence ψ φ ψ 1 φ z = 2 = 2 , z = 3 = x ix (ix + x ) 2 , 1 ψ φ 2 ψ 1+ x 4 − 3 − 1 2 φ 1 1 1 5 1 ψ 1 φ z = 4 = ix + x + (x + ix ) 2 , 3 ψ 1+ x − 1 2 4 3 φ 1 5 1 and also for the LLL basis elements, (5.13) and (B.5) for k = 3.

Note added After completion of this work, the author learned the works [101, 102]. In the appendices of the 2 3 4 8 papers, the fuzzy spheres (SF , SF , SF and SF ) are constructed in the context of embedding in Moyal spaces, with emphasis on relations to the Hopf maps. Such constructions are completely consistent with the descriptions in the present paper. The author is grateful to Mohammad M. Sheikh-Jabbari for the information.

Acknowledgements I would like to thank Yusuke Kimura for collaborations. Many crucial ingredients in this review are based on the works with him. I am also indebted to Takehiro Azuma for email correspondence about mathematics of fuzzy spheres. Since this article is a review-type, many important works not performed by the author are included. Hereby, I express my gratitude to the researchers whose works are reviewed in the paper.

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